Efficient modular multiplication algorithms for public key cryptography

The modular exponentiation is an important operation for cryptographic transformations in public key cryptosystems like the Rivest, Shamir and Adleman, the Difie and Hellman and the ElGamal schemes. computing a
^x
mod n and a
^x
b
^y
mod n for very large x,y and n are fundamental to the efficiency of almost all pubic key cryptosystems and digital signature schemes. To achieve high level of security, the word length in the modular exponentiations should be significantly large. The performance of public key cryptography is primarily determined by the implementation efficiency of the modular multiplication and exponentiation. As the words are usually large, and in order to optimize the time taken by these operations, it is essential to minimize the number of modular multiplications. In this paper we are presenting efficient algorithms for computing a
^x
mod n and a
^x
b
^y
mod n. In this work we propose four algorithms to evaluate modular exponentiation. Bit forwarding (BFW) algorithms to compute a
^x
mod n, and to compute a
^x
b
^y
mod n two algorithms namely Substitute and reward (SRW), Store and forward(SFW) are proposed. All the proposed algorithms are efficient in terms of time and at the same time demands only minimal additional space to store the pre-computed values. These algorithms are suitable for devices with low computational power and limited storage.